To determine which set of values could be the side lengths of a [tex]\(30^\circ-60^\circ-90^\circ\)[/tex] triangle, we need to examine the properties of this special type of right triangle.
In a [tex]\(30^\circ-60^\circ-90^\circ\)[/tex] triangle:
- The length of the hypotenuse (the side opposite the [tex]\(90^\circ\)[/tex] angle) is twice the length of the shorter leg (the side opposite the [tex]\(30^\circ\)[/tex] angle).
- The length of the longer leg (the side opposite the [tex]\(60^\circ\)[/tex] angle) is [tex]\(\sqrt{3}\)[/tex] times the length of the shorter leg.
Given these properties, we need to check each option to see if these conditions are met.
Option A: [tex]\(\{5, 10, 10\sqrt{3}\}\)[/tex]
- Shorter leg = 5
- Hypotenuse = 10
- Longer leg = 10[tex]\(\sqrt{3}\)[/tex]
Let's check the conditions:
- Hypotenuse should be twice the shorter leg: [tex]\(2 \times 5 = 10\)[/tex]. This is true.
- Longer leg should be [tex]\(\sqrt{3}\)[/tex] times the shorter leg: [tex]\(5 \times \sqrt{3} = 5\sqrt{3}\)[/tex]. However, the given longer leg is 10[tex]\(\sqrt{3}\)[/tex], which is not correct.
Option B: [tex]\(\{5, 10, 10\sqrt{2}\}\)[/tex]
- Shorter leg = 5
- Hypotenuse = 10
- Longer leg = 10[tex]\(\sqrt{2}\)[/tex]
Let's check the conditions:
- Hypotenuse should be twice the shorter leg: [tex]\(2 \times 5 = 10\)[/tex]. This is true.
- Longer leg should be [tex]\(\sqrt{3}\)[/tex] times the shorter leg: [tex]\(5 \times \sqrt{3} = 5\sqrt{3}\)[/tex]. However, the given longer leg is [tex]\(10\sqrt{2}\)[/tex], which is not correct.
Option C: [tex]\(\{5, 5\sqrt{2}, 10\}\)[/tex]
- Shorter leg = 5
- Hypotenuse = 10
- Longer leg = 5[tex]\(\sqrt{2}\)[/tex]
Let's check the conditions:
- Hypotenuse should be twice the shorter leg: [tex]\(2 \times 5 = 10\)[/tex]. This is true.
- Longer leg should be [tex]\(\sqrt{3}\)[/tex] times the shorter leg: [tex]\(5 \times \sqrt{3} = 5\sqrt{3}\)[/tex]. However, the given longer leg is [tex]\(5\sqrt{2}\)[/tex], which is not correct.
Option D: [tex]\(\{5, 5\sqrt{3}, 10\}\)[/tex]
- Shorter leg = 5
- Hypotenuse = 10
- Longer leg = 5[tex]\(\sqrt{3}\)[/tex]
Let's check the conditions:
- Hypotenuse should be twice the shorter leg: [tex]\(2 \times 5 = 10\)[/tex]. This is true.
- Longer leg should be [tex]\(\sqrt{3}\)[/tex] times the shorter leg: [tex]\(5 \times \sqrt{3} = 5\sqrt{3}\)[/tex]. This is true.
After examining all the options, we see that the correct set of values that correspond to the side lengths of a [tex]\(30^\circ-60^\circ-90^\circ\)[/tex] triangle is:
D. [tex]\(\{5, 5\sqrt{3}, 10\}\)[/tex]
